This problem is a result of non-management. Perhaps some management would rectify the situation.
Everyone else is exploring the "PIP and fire them now" side. I'm going to explore other possibilities that may allow you to avoid firings and avoid good people leaving.
Now, perhaps you wonder why I don't fire them. We cannot find other
workers at the time to take their place, and we have a lot of orders
People that are hard to replace typically require a lot of training or education to do their jobs. Unskilled laborers are easier to replace than skilled laborers.
Therefore, it sounds like you have knowledge workers of some type.
If you follow the harsh recommendations I'm seeing here, I think they'll be very unhappy and much more likely to leave. You could destroy your own business if they feel you've mistreated them.
[they] do not allow newer workers hired in the same department to integrate into the team
If they are knowledge workers, perhaps they have good reason to keep the new guys at arm's length. It could be that they have a long-term strategy to integrate the new people and you simply aren't aware of it yet. It could be a short-sighted strategy based on pressure to deliver. It could be that they really aren't cooperating with your strategy of making them more easily redundant. But since your business relies on them and they are so difficult to replace, you need to take a more careful tack with them than firings and warnings.
Perhaps you could better use their self-interested reaction to pressures for your benefit and thus better align everyone's interests. Some suggestions along these lines:
Talk to them about their strategy to integrate the new hires. Find out what their concerns are about the new people. Ask if they've created any problems. See if you can do anything to help (you probably can't, but the attempt shows that you're on the same side of the table as they are). Maybe the new people need training that the more tenured folk haven't made time for. Give them a balanced scorecard so that they can make the time to do these activities without feeling like they're sacrificing their performance goals.
Elevate those with the most leadership potential to management, and make them responsible for getting productivity out of the new people. Management isn't for everyone, if you determine a leader you picked isn't working out, you'll want to move them back into the single contributor role while feeling that their compensation is handled fairly. You'll want to structure their compensation to separately compensate for their management responsibilities. Maybe you'll have several managers, each in charge of integrating a few people.
Hire a manager from the outside to be solely responsible for the above strategies and for optimizing your long-term output. He'll be much easier to fire if he's not working out.
From the original question:
The tension has escalated to the point that some of the new hires have resigned.
A commenter comments:
This makes them (at the very best) extremely bad at managing people, and the suggestion that you promote any of the to a management position is laughable.
I think that's a simplistic view. In fact, we don't know very much about this situation. If they're knowledge workers, as I am assuming, there may well be a few of them that would make good managers. Maybe there are others at the firm that could fill that role. Maybe it's impossible, and the recommendation (one of several) is no good. But if your people leave and you can't replace them, you'll be out of business.
Minimum wage jobs can be harder to fill if other unskilled positions pay more.
If what you're assuming is true, the simple solution, then, is to pay a competitive wage.
If they can't pay competitively, then the firm's business model is built on an unsustainable strategy of only paying below-market wages.
If the firm can't afford to pay competitive wages, it's a question of its competitiveness in the industry. If it can't compete then it should fail.
Let's imagine, on the other hand, that this is a software development shop. Stealing from production was, for example, someone running an IRC client on a production server (which they shouldn't do, but maybe they felt entitled to as a job perk or as part of the culture, or maybe they're actually using it for business purposes). Then these are developers, and the senior developers naturally want to keep their new hires at arms length until they learn the processes and expectations of their development culture. If that's the case, here's the other side of this question: What is a 'friendly' way to let managers know that having good developers is a privilege?